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    Table of Integrals, Series, and Products
    Copyright page
    Contents
    Preface to the Seventh Edition
    Acknowledgments
    The Order of Presentation of the Formulas
    Use of the Tables
    Index of Special Functions
    Notation
    Note on the Bibliographic References
    Chapter 0 Introduction
    0.1 Finite Sums
    0.2 Numerical Series and Infinite Products
    0.3 Functional Series
    0.4 Certain Formulas from Differential Calculus
    Chapter 1 Elementary Functions
    1.1 Power of Binomials
    1.2 The Exponential Function
    1.3-1.4 Trigonometric and Hyperbolic Functions
    1.5 The Logarithm
    1.6 The Inverse Trigonometric and Hyperbolic Functions
    Chapter 2 Indefinite Integrals of Elementary Functions
    2.0 Introduction
    2.1 Rational Functions
    2.2 Algebraic Functions
    2.3 The Exponential Function
    2.4 Hyperbolic Functions
    2.5-2.6 Trigonometric Functions
    2.7 Logarithms and Inverse-Hyperbolic Functions
    2.8 Inverse Trigonometric Functions
    Chapter 3-4 Definite Integrals of Elementary Functions
    3.0 Introduction
    3.1-3.2 Power and Algebraic Functions
    3.3-3.4 Exponential Functions
    3.5 Hyperbolic Functions
    3.6-4.1 Trigonometric Functions
    4.2-4.4 Logarithmic Functions
    4.5 Inverse Trigonometric Functions
    4.6 Multiple Integrals
    Chapter 5 Indefinite Integrals of Special Functions
    5.1 Elliptic Integrals and Functions
    5.2 The Exponential Integral Function
    5.3 The Sine Integral and the Cosine Integral
    5.4 The Probability Integral and Fresnel Integrals
    5.5 Bessel Functions
    Chapter 6-7 Definite Integrals of Special Functions
    6.1 Elliptic Integrals and Functions
    6.2-6.3 The Exponential Integral Function and Functions Generated by It
    6.4 The Gamma Function and Functions Generated by It
    6.5-6.7 Bessel Functions
    6.8 Functions Generated by Bessel Functions
    6.9 Mathieu Functions
    7.1-7.2 Associated Legendre Functions
    7.3-7.4 Orthogonal Polynomials
    7.5 Hypergeometric Functions
    7.6 Confluent Hypergeometric Functions
    7.7 Parabolic Cylinder Functions
    7.8 Meijer's and MacRobert's Functions (G and E)
    Chapter 8-9 Special Functions
    8.1 Elliptic Integrals and Functions
    8.2 The Exponential Integral Function and Functions Generated by It
    8.3 Euler's Integrals of the First and Second Kinds
    8.4-8.5 Bessel Functions and Functions Associated with Them
    8.6 Mathieu Functions
    8.7-8.8 Associated Legendre Functions
    8.9 Orthogonal Polynomials
    9.1 Hypergeometric Functions
    9.2 Confluent Hypergeometric Functions
    9.3 Meijer's G-Function
    9.4 MacRobert's E-Function
    9.5 Riemann's Zeta Functions zeta(z,q) and zeta(z), and the Functions Phi(z,s,v) and xi(s)
    9.6 Bernoulli Numbers and Polynomials, Euler Numbers
    9.7 Constants
    Chapter 10 Vector Field Theory
    10.1-10.8 Vectors, Vector Operators, and Integral Theorems
    Chapter 11 Algebraic Inequalities
    11.1-11.3 General Algebraic Inequalities
    Chapter 12 Integral Inequalities
    12.11 Mean Value Theorems
    12.21 Differentiation of Definite Integral Containing a Parameter
    12.31 Integral Inequalities
    12.41 Convexity and Jensen's Inequality
    12.51 Fourier Series and Related Inequalities
    Chapter 13 Matrices and Related Results
    13.11-13.12 Special Matrices
    13.21 Quadratic Forms
    13.31 Differentiation of Matrices
    13.41 The Matrix Exponential
    Chapter 14 Determinants
    14.11 Expansion of Second- and Third-Order Determinants
    14.12 Basic Properties
    14.13 Minors and Cofactors of a Determinant
    14.14 Principal Minors
    14.15* Laplace Expansion of a Determinant
    14.16 Jacobi's Theorem
    14.17 Hadamard's Theorem
    14.18 Hadamard's Inequality
    14.21 Cramer's Rule
    14.31 Some Special Determinants
    Chapter 15 Norms
    15.1-15.9 Vector Norms
    15.11 General Properties
    15.21 Principal Vector Norms
    15.31 Matrix Norms
    15.41 Principal Natural Norms
    15.51 Spectral Radius of a Square Matrix
    15.61 Inequalities Involving Eigenvalues of Matrices
    15.71 Inequalities for the Characteristic Polynomial
    15.81-15.82 Named Theorems on Eigenvalues
    15.91 Variational Principles
    Chapter 16 Ordinary Differential Equations
    16.1-16.9 Results Relating to the Solution of Ordinary Differential Equations
    16.11 First-Order Equations
    16.21 Fundamental Inequalities and Related Results
    16.31 First-Order Systems
    16.41 Some Special Types of Elementary Differential Equations
    16.51 Second-Order Equations
    16.61-16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations
    16.71 Two Related Comparison Theorems
    16.81-16.82 Non-Oscillatory Solutions
    16.91 Some Growth Estimates for Solutions of Second-Order Equations
    16.92 Boundedness Theorems
    Chapter 17 Fourier, Laplace, and Mellin Transforms
    17.1-17.4 Integral Transforms
    Chapter 18 The z-Transform
    18.1-18.3 Definition, Bilateral, and Unilateral z-Transforms
    References
    Supplemental references
    Index of Functions and Constants
    General Index of Concepts
    Table of Integrals, Series, and Products Seventh Edition
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    Table of Integrals, Series, and Products Seventh Edition I.S. Gradshteyn and I.M. Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Translated from Russian by Scripta Technica, Inc. AMSTERDAM • BOSTON • HEIDELBERG • LONDON SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO NEW YORK • OXFORD • PARIS • SAN DIEGO Academic Press is an imprint of Elsevier
    Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. ∞ Copyright c 2007, Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.com. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” For information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com ISBN-13: 978-0-12-373637-6 ISBN-10: 0-12-373637-4 PRINTED IN THE UNITED STATES OF AMERICA 07 08 09 10 11 9 8 7 6 5 4 3 2 1
    Contents Preface to the Seventh Edition Acknowledgments The Order of Presentation of the Formulas Use of the Tables Index of Special Functions Notation Note on the Bibliographic References 0 0.1 0.2 0.3 0.4 1 1.1 1.2 Introduction 0.11 0.12 0.13 0.14 0.15 0.21 0.22 0.23–0.24 0.25 0.26 0.30 0.31 0.32 0.33 0.41 0.42 0.43 0.44 Finite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions Sums of powers of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . Sums of reciprocals of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sums of products of reciprocals of natural numbers Sums of the binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Series and Infinite Products . . . . . . . . . . . . . . . . . . . . . . . The convergence of numerical series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of numerical series Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of infinite products Functional Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power series Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Certain Formulas from Differential Calculus . . . . . . . . . . . . . . . . . . . . Differentiation of a definite integral with respect to a parameter . . . . . . . . . The nth derivative of a product (Leibniz’s rule) . . . . . . . . . . . . . . . . . . The nth derivative of a composite function . . . . . . . . . . . . . . . . . . . . Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Functions 1.11 1.12 Power of Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power series Series of rational fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v xxi xxiii xxvii xxxi xxxix xliii xlvii 1 1 1 1 3 3 3 6 6 6 8 14 14 15 15 16 19 21 21 21 22 22 23 25 25 25 26 26
    vi CONTENTS 1.3–1.4 1.21 1.22 1.23 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.41 1.42 1.43 1.44–1.45 1.46 1.47 1.48 1.49 1.51 1.52 1.61 1.62–1.63 1.64 2.00 2.01 2.02 2.10 2.11–2.13 2.14 2.15 2.16 2.17 2.18 2.20 2.21 1.5 1.6 2 2.0 2.1 2.2 Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of exponentials . . . . . . . . . . . . . . . . . . . . . Trigonometric and Hyperbolic Functions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The basic functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) . . . . . . . . . . . . . . . . . The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions . . . . . . . . . . . . . . . . . . . . . . . . Certain sums of trigonometric and hyperbolic functions Sums of powers of trigonometric functions of multiple angles . . . . . . . . . . . . . . . . . . . Sums of products of trigonometric functions of multiple angles Sums of tangents of multiple angles . . . . . . . . . . . . . . . . . . . . . . . . Sums leading to hyperbolic tangents and cotangents . . . . . . . . . . . . . . . The representation of cosines and sines of multiples of the angle as finite products . . . . The expansion of trigonometric and hyperbolic functions in power series . . . . . . . . . . . . . . . . . . . . . . Expansion in series of simple fractions Representation in the form of an infinite product . . . . . . . . . . . . . . . . . Trigonometric (Fourier) series . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of products of exponential and trigonometric functions . . . . . . . . . . Series of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Lobachevskiy’s “Angle of Parallelism” Π(x) . . . . . . . . . . . . . . . . . . . The hyperbolic amplitude (the Gudermannian) gd x . . . . . . . . . . . . . . . The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series of logarithms (cf. 1.431) The Inverse Trigonometric and Hyperbolic Functions . . . . . . . . . . . . . . . The domain of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Series representations Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General remarks The basic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing the binomial a + bxk . . . . . . . . . . . . . . . . . . . . . . Forms containing the binomial 1 ± xn . . . . . . . . . . . . . . . . . . . . . . Forms containing pairs of binomials: a + bx and α + βx . . . . . . . . . . . . . Forms containing the trinomial a + bxk + cx2k . . . . . . . . . . . . . . . . . . Forms containing the quadratic trinomial a + bx + cx2 and powers of x . . . . Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Functions √ Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . . Forms containing the binomial a + bxk and 26 27 27 28 28 28 31 33 36 37 38 39 39 41 42 44 45 46 51 51 51 52 53 53 55 56 56 56 60 63 63 63 64 65 66 66 68 74 78 78 79 81 82 82 83 Indefinite Integrals of Elementary Functions
    CONTENTS 2.22–2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.31 2.32 2.41–2.43 2.44–2.45 2.46 2.47 2.48 2.50 2.51–2.52 2.53–2.54 2.3 2.4 2.5–2.6 2.55–2.56 2.57 2.58–2.62 2.63–2.65 2.66 2.67 2.71 2.72–2.73 2.74 2.81 2.82 2.83 2.84 2.85 2.7 2.8 √ (a + bx)k . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing n √ a + bx and the binomial α + βx . . . . . . . . . . . . . . . Forms containing √ a + bx + cx2 . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing √ a + bx + cx2 and integral powers of x . . . . . . . . . . . . Forms containing √ a + cx2 and integral powers of x . . . . . . . . . . . . . . . Forms containing a + bx + cx2 and first- and second-degree polynomials . . . Forms containing . . . . . . . Integrals that can be reduced to elliptic or pseudo-elliptic integrals The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forms containing eax The exponential combined with rational functions of x . . . . . . . . . . . . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of sinh x, cosh x, tanh x, and coth x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational functions of hyperbolic functions Algebraic functions of hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of hyperbolic functions and powers Combinations of hyperbolic functions, exponentials, and powers . . . . . . . . . Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers of trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . Sines and cosines of multiple angles and of linear and more complicated func- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tions of the argument √ √ Rational functions of the sine and cosine . . . . . . . . . . . . . . . . . . . . . a ± b sin x or a ± b cos x . . . . . . . . . . . . . . . . . Integrals containing . . . . . . . . . . . . Integrals reducible to elliptic and pseudo-elliptic integrals . . . . . . . . . . . . . . . . . Products of trigonometric functions and powers Combinations of trigonometric functions and exponentials . . . . . . . . . . . . Combinations of trigonometric and hyperbolic functions . . . . . . . . . . . . . Logarithms and Inverse-Hyperbolic Functions . . . . . . . . . . . . . . . . . . . The logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of logarithms and algebraic functions Inverse hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arcsines and arccosines . . . . . The arcsecant, the arccosecant, the arctangent, and the arccotangent Combinations of arcsine or arccosine and algebraic functions . . . . . . . . . . . Combinations of the arcsecant and arccosecant with powers of x . . . . . . . . Combinations of the arctangent and arccotangent with algebraic functions . . . 3–4 Definite Integrals of Elementary Functions 3.0 3.01 3.02 3.03 3.04 3.05 3.11 3.1–3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorems of a general nature . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of variable in a definite integral . . . . . . . . . . . . . . . . . . . . . . General formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The principal values of improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power and Algebraic Functions Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 84 88 92 94 99 103 104 106 106 106 110 110 125 132 139 148 151 151 151 161 171 179 184 214 227 231 237 237 238 240 241 241 242 242 244 244 247 247 247 248 249 251 252 253 253
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